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Boundary conditions associated with the general left-definite theory for differential operators
Stockholm University, Faculty of Science, Department of Mathematics.
Number of Authors: 32019 (English)In: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 239, p. 1-28Article in journal (Refereed) Published
Abstract [en]

In the early 2000s, Littlejohn and Wellman developed so-called nth left-definite theory. Namely, they fully determined the 'left-definite domains' and spectral properties of powers of self-adjoint Sturm-Liouville operators associated with classical orthogonal polynomials. We study how these left-definite domains relate with explicit classical Glazman-Krein-Naimark (GKN) boundary conditions. When n is small, we significantly simplify previously challenging analysis by introducing an explicit method for checking whether a given set of functions yields GKN conditions. This reduces to computing the rank of a relatively small matrix. We include explicit computations for n = 2, ... , 5. Further, for arbitrary powers n of Sturm-Liouville operators with a complete system of orthogonal eigenfunctions, we show that these left-definite domains are given by GKN boundary conditions involving some of the polynomial eigenfunctions. We also study and extend a conjecture by Littlejohn-Wicks regarding the equality of four different formulations for these domains.

Place, publisher, year, edition, pages
2019. Vol. 239, p. 1-28
Keywords [en]
Orthogonal polynomials, Left-definite theory, Sturm-Liouville operators, Glazman-Krein-Naimark theory, Boundary conditions
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-167652DOI: 10.1016/j.jat.2018.10.005ISI: 000459368700001OAI: oai:DiVA.org:su-167652DiVA, id: diva2:1302466
Available from: 2019-04-04 Created: 2019-04-04 Last updated: 2019-04-04Bibliographically approved

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